The Discontinuity Point Sets of Quasi-Continuous Functions

BULL. AUSTRAL. MATH. SOC. 54C30, 54clO VOL. 75 (2005) [373-379]

THE DISCONTINUITY POINT SETS OF QUASI-CONTINUOUS FUNCTIONS

OLEKSANDR V. MASLYUCHENKO

It is proved that a subset E of a hereditarily normal topological space X is a discontinuity point set of some quasi-continuous function / : X -¥ K if and only if E is a countable union of sets En = ~An f~l Bn where 3nnBI1 = J4rnln = 0.

1. INTRODUCTION We deal with an old problem of the construction of a function with a given discontinuity point set. Separately continuous functions with given discontinuity point sets were constructed in [6, 5, 1, 8]. A complete characterisation of discontinuity point sets for separately continuous functions defined on metrisable spaces was obtained in [9]. An exact problem on construction of functions with given oscillations was considered in [7, 2, 4, 10, 11, 12]. Note that the domain space of all functions was assumed to be metrisable or "near" metrisable in all above papers. In our note we characterise discontinuity point sets of quasi-continuous functions denned on a hereditarily normal space. We prove in Theorem 4.2 that a subset E of a hereditarily normal topological space X is a discontinuity point set of some quasi- continuous function / : X ->• R if and only if E is a a-junction set (that is, E is a countable

union of sets En - ~An n Sn where InnBn = innBn = 0). In [12, Theorem 2.11.1] it was proved that a function tp : X -> [0, +oo] is the oscillation of some quasi-continuous function / : X -t R if and only if the sets tp~l([e, +oo]) are closed nowhere dense, for each e > 0. But it is easy to see that for any meager F^-set E C. X there is a function ip : X -)• [0, +oo] such that E = ip'1 ({0}) and the sets 0. So, a subset E of a metrisable space X is a discontinuity point set of some quasi-continuous function / : X -¥ R if and

only if E is a meager Fa-set. But this fact may to be implied also from our the main result (Theorem 4.2). Indeed, Theorem 2.4 yields that in the metrisable case the exjunction set is exactly the meager

Fa-set. Moreover, in Theorem 4.3 we obtain that if A" is a perfectly normal hereditary quasi- separable (or simple, hereditary separable) Frechet-Uryson space then a subset E of X Received 16th October, 2006 Copyright Clearance Centre, Inc. Serial-fee code: 0004-9727/05 SA2.00+0.00.

373

Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.8, on 24 Sep 2021 at 11:18:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700039307 374 O.V. Maslyuchenko [2]

is a discontinuity point set of some quasi-continuous function / : X —> R if and only if

E is a meager Fff-set.

2. JUNCTION SETS A subset E of a topological space X we call a junction set if there are subsets A and B of X such that ~AC\B = AC\~B = 0 and £ = InB. PROPOSITION 2.1. Let £ be a junction set of a topological space X. Then E is closed nowhere dense in X.

PROOF: Let A and B be subsets of X such that ~AV\B = ACYB = 0 and E = A~C\~B. Then obviously E is closed. Thus for nowhere density of E it is enough to show that int.E = 0. Since intInBCAnB = 0 then int 4 n 5 = 0. Hence intE = int~Anint23 CintlnB = 0. D Under some additional assumptions on X (say, X is metrisable) we shall prove that the junction sets are exactly the closed nowhere dense sets.

Recall some definitions. A family (Ai)i€l of subsets of a topological space X is said to be discrete (locally finite) if any point of X has a neighbourhood which intersects with at most one (a finite number) of Ai's. A system A C 2X is calls a-locally finite if A is a countable union of locally finite systems. A system A is said to be a net if for any point x G X and neighbourhood U of x there is A G A with x G A C U. A subset

S C X we call strongly discrete if there exists an discrete family of open sets Us, s & S, such that s G U, for each s £ S. It is easy to see that any closed discrete subset of a paracompact space is strongly discrete. A subset S C X will be called strongly a-discrete if 5 is a countable union of strongly discrete sets. A subset fiCIwe shall call strongly a-discrete if it contains a dense strongly a-discrete subset. A topological space X will be called hereditarily quasi-separable if every subset of X is strongly a-discrete in X. PROPOSITION 2.2. Let X be a paracompact space with a a-locally finite net A. Then X is hereditary quasi-separable.

PROOF: Suppose E C.X. We set AE = {A G A : A n E = 0} and for each A e AE

choose an sA G A D E. Since A is a net then the set 5 = {SA '• A G AE} is dense in E. By a-locally finiteness of A we obtain that 5 is a countable union of closed discrete sets. Hence 5 is strongly a-discrete, because X is paracompact. Thus, E is strongly a-discrete. D Recall that a topological space X is said to be a Frechet-Uryson space if for any

AC. X and x £ A there exists a sequence of points xn G A such that xn —¥ x. An E C X oo is called a Gs-set if there is a sequence of open sets GnD E with E = f) Gn THEOREM 2.3. Let X be a Frechet-Uryson space and E be a closed nowhere dense strongly a-discrete Gs-subset of X. Then E is a junction set in X.

PROOF: Let 5 be a dense strongly a-discrete subset of E and Sn be strongly discrete

sets with S = \J Sn. Without lost of generality we may assume that Sn are disjoint. Fix n=l oo a decreasing sequence of open subsets GnD E with E = f] Gn. For arbitrary n € N pick n=l

an open discrete family (£/(s) : s G S) so that s G U(s) C Gn for s G Sn. We construct a

family (tk(s) : k G N, s G 5) so that for arbitrary fc, k', k" G N and s, s', s" G 5 we have (1) tk(s)eU\E;

(2) tfc-(s')#tfc»(s")if(^s')^(A:",S");

(3) tk(s) —y s as k —• oo.

Suppose that tjt(s) are already constructed for some m € N and each s € \J Sn

and A e N so that (l)-(3) hold. Let us construct tk(s) for s e Sm and A; e N. Put

Tn = {tk(s) : k £N,s £ Sn} for n < m. Since the family (U(s) : s G 5n) is discrete and

•5n n Sm = 0 for n < m then (1) and (3) imply that Sm n Tn = 0 for n < m. Now fix

s € 5m. Since A" is a Frechet-Uryson space and

s G U{s) n

then there is a sequence tk(s) G U(s) \ (E U (J Tn) such that tk{s) -*• 5. Obviously, (l)-(3) are valid. Now set

An = {t2k-i(s) :k£N,s£Sn},

Bn = {t2k(s):kef!,s<=Sn}, 00 A = [J An and

oo B=\jBn. n=\

Since (t/(s) : s G 5n) are discrete then An = An(l Sn and Bn = Bn n 5n. Taking into

account that U(s) C Gn for s G Sn we obtain that An, Bn C Gn. Then for each m 6 N one has

n>m n%m n

= S=\JSnc{jAn = A. n=l n=l

Hence A = AUE. Analogously, B = BuE. Thus, ~Af\B = An~B = 0 and E = An~B. Therefore E is a junction set. D By the above results we deduce the following. THEOREM 2.4. Suppose that for a Frechet-Urysoa space X at least, one of the following conditions holds (i) X is a hereditarily separable perfectly normal; (ii) X is a hereditarily quasi-separable perfectly normal; (iii) X is a regular space with a countable net; (iv) X is a paracompact with a a-locally finite net; (v) X is metrisable. Then a subset E ofX is a junction set in X if and only ifE is closed nowhere dense. Remark that the assumption A" be a Frechet-Uryson space is essential in the above two theorems. Indeed, consider the subspace X = N U {

3. A PROOF THAT THE DISCONTINUITY POINT SET OF A QUASI-CONTINUOUS FUNCTION IS A a-JUNCTION SET

We recall that a function / : X -> Y is called quasi-continuous if f~1(G) C intf^iG) for each open G C Y. A subset E of a topological space X we call a oo a-junction set if there is a sequence of junction sets En with E = (J En. 71=1 THEOREM 3.1. Let X be a topological space, Y be a separable metrisable space and f : X —• Y be a quasi-continuous function. Then the discontinuity point set D(f) of the function f is a-junction set.

PROOF: Let V be a countable base of Y. Consider a countable system

{{V, W) € V x V : V C W} = {(Vn, Wn) : n G N}. l Set An = intf- {Vn) and Bn = int f-\Y \ Wn). Since f{An) C Vn and f{Bn)

C Y \Wn C Y\Vn then AnC\Bn = 0. Taking into account that An and Bn are

open one obtains An l~l Bn = An n Bn = 0. Thus, En = An D Bn are junction sets.

Besides, f(An) n /(£„) = 0 since Fn C Wn. Therefore En C £)(/). It is left to prove that D(f) C 0 ^n- n=l tnen Given io € ^(/)> there exists a neighbourhood V of /(x

Since V is a base of Y then there is n € N such that /(i0) e Vn and

l Wn C V. But / is quasi-continuous. Then f~ {Vn) Q An and

r\Y\V)Cf-\Y\Wn)CA-n. Thus, l l so € f~ (Vn) D f~ (Y\V) C.A~nC\~Bn = En oo and we obtain £>(/) C \J En. U n=l

4. A CONSTRUCTION OF A QUASI-CONTINUOUS FUNCTION WITH A GIVEN DISCONTINUITY POINT SET Recall that a function / : X —> R = [—oo, +00] is said to be upper (lower) semi- continuous if the set /~l([—00, y)) (respectively, /^((y, +00])) is open for each y € R. By an upper (lower) Baire function of/ we understand the function /* (respectively, /,) defined by

f'(x) = inf supf(U) and f.(x) = sup inf/(t/) for x e X.

An upper (lower) Baire function is upper (lower) semi-continuous. PROPOSITION 4.1. Let X be a topological spaces and f : X ->• R be an iower (upper) semi-continuous function. Then /* (respectively, /.j is quasi-continuous. PROOF: Let / be, say, lower semi-continuous. Since /* is upper semi-continuous then it remains to prove that /* is lower quasi-continuous (that is, for given XQ €. X,

7/o < f*(xo) and a neighbourhood UQ of xo there is an open nonempty set U C Uo such that /*(x) > j/o when x € U). Suppose XQ G X, y0 < f*(x0) and let £/0 be an open neighbourhood of xo- Since j/o < /*(^o) ^ supf(Uo) then there is an ii e f/o with /(xi) > j/o- But / is lower semi-continuous. Then there exists an open neighbourhood

U C Uo of xi such that /(x) > y0 on [/. Finally, f*(x) ^ /(x) > j/0 on U. D THEOREM 4.2. Let X be a hereditarily normal space and E C X. Then E is a discontinuity point set of some quasi-continuous function f : X -» R if and only if E is a a-junction set.

PROOF: Let En be junction sets such that E = \J En. Choose sets An and Bn _ _ _n=l such that AnnBn = AnDBn =

normal, An D En = Bn C\ En = 0 and (/!„ 0 Bn) \ En = 0 then by the Uryson lemma

[3] there exists a continuous function /„ : X \ En -4 [0,1] such that fn(x) = 0 on j4n

and Jn(x) = 1 on Bn. Define /n(x) = 0 for x € £„ and fn(x) = Jn(x) for x 6 X \ £•„.

Since f(An) nf(Bn) = 0 and En = 34nDSn then Fn C JD(/n). But /U\£n is continuous.

Thus £„ = D(fn). Besides, the points x e En = D(fn) are the minimum points of /„. Therefore /„ is lower semi-continuous.

oo n Define g = £ (l/4 )/n and / = g*. Since g is lower semi-continuous then / is quasi- n=l continuous by Proposition 4.1. Let us prove that £>(/) = E. First, if all /„ are continuous oo at a given point x0 € X then so is g, and hence /. Thus, E = \J D(fn) D D(f). n=l

Given x0 € E, we show that x0 € D(f). Let n be the least integer such that

x0 € En. We set e = 1/4", u = £ (1/4*)/* and v = £ (1/4*)/*. Since x0 ? Ek = D(fk) kn

for A; < n then u is continuous at x0. Therefore there is an open neighbourhood U of 1 x0 such that \u(x) - u(xo)\ < e/9 on for x € U. Put G = U n int/" ^, 1/9)) and 1 # = £/ n int Z" ((8/9,1]). Since /„ is continuous on X \ En D An, Bn and fn(x) = 0 on An and f (x) = 1 on B then .4 DC/ C G and B ("IU C i/. Hence x € Gnff. Besides, n n n +1n 0 4" _ 1 _ £ *>„- -~^W " ^ " 3 for x E X. Thus for x 6 G one has

ff(x) = u(x) + e/n(x) + v(x) < u(x0) + - + - + - = u(x0) + —.

For y € H we obtain the converse inequality

9{y) ^ u(j/) + efn(y) > u(x0) - - + — = u(x0) + —. y y y

But since f = g* and the sets G and ff are open then for any x 6 G and y & H

f(x) ^ ti(xo) + y < u(x0) + y ^ /(y).

From i0 6 G n H we finally obtain that / is discontinuous at x0. D Theorems 2.4 and 4.2 together imply the following. THEOREM 4.3. Let for a Frechet-Uryson space X at least, one of the following conditions holds (i) X is a hereditarily separable perfectly normal; (ii) X is a hereditarily quasi-separable perfectly normal; (iii) X is a regular space with a countable net; (iv) X is a paracompact with a o-locally finite net; (v) X is metrisable. Then a subset E of X is the discontinuity point sets of some quasi-continuous func-

tion f : X —> R if and only if E is a meager Fa-set.

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Department of Mathematic Jurij Fed'kovych Chernivtsi National University vul Kotsyubyns'kolio 2 Cherniutsi 58012 Ukraine e-mail: [email protected]